TITE DTP Scoping
Amit Patel
23-Jun-2021
1 Example Trial
prior <- c(0.1, 0.2, 0.3)
target <- 0.2
obswin <- 56Consider a trial with:
- 3 dose levels
- A target DLT rate of 20% (TD20)
- Prior guess of TD20 at dose level 2
- Follow up period of 8 weeks (56 days)
- Patients starting at dose level 1
- A linear weighting
- TITE-CRM using the empiric model
2 Simple Case
The simplest case would involve just a single patient. There are two possible outcomes:
- Toxicity (T)
- No toxicity (N)
2.1 If Toxicity Occurs
If a toxicity occurs, the time it happens is irrelevent as the weighting is automatically set at 1,
level <- 1
tox <- 1
followup <- 56
mod <- titecrm(prior = prior, target = target, tox = tox, level = level,
obswin = obswin, scheme = 'linear', followup = followup)
mod## Today: Wed Jun 23 11:18:58 2021
## DATA SUMMARY (TITE-CRM)
## PID Level Toxicity f/u Weight Included
## 1 1 1 56 1 1
##
## Toxicity probability update (with 90 percent probability interval):
## Level Prior n total.wts total.tox Ptox LoLmt UpLmt
## 1 0.1 1 1 1 0.507 0.078 0.835
## 2 0.2 0 0 0 0.622 0.168 0.882
## 3 0.3 0 0 0 0.701 0.263 0.91
## Next recommended dose level: 1
## Recommendation is based on a target toxicity probability of 0.2
##
## Estimation details:
## Empiric dose-toxicity model: p = dose^{exp(beta)}
## dose = 0.1 0.2 0.3
## Normal prior on beta with mean 0 and variance 1.34
## Posterior mean of beta: -1.222
## Posterior variance of beta: 0.65The reccommended dose is dose level 1.
2.2 If No Toxicity Occurs
If no toxicity is observed the next dose allocated would depend on the follow up of the patient. Here there would be 56 different possibilities. The patient could be followed up for either \(1, 2, 3, ..., 56\) days.
tox <- 0
followup <- 1:obswin
results <- data.frame('Observed' = followup, 'Rec' = rep(0,obswin))
for (i in 1:obswin) {
followup <- i
mod <- titecrm(prior = prior, target = target, tox = tox, level = level,
obswin = obswin, scheme = 'linear', followup = followup)
results$Rec[i] <- mod$mtd
}We can determine the next dose by looping through all possible values of follow up. The results can be summarised using the min and max follow up days for each dose.
results <- results %>% group_by('Next Dose' = Rec) %>%
summarise('Follow up data' = paste(min(Observed), max(Observed),
sep ='-'))
kable(results, align = 'cc') %>%
kable_styling(full_width = F, position = 'left')| Next Dose | Follow up data |
|---|---|
| 2 | 1-30 |
| 3 | 31-56 |
This can be interpreted as the next dose will be 2 if the previous patien has no toxicity and has been observed between 1 and 30 days. Likewise, the next dose level would be 3 if no toxicities were observed between days 31 and 56.
2.3 DTP for 1 Patient
kable(results %>%
pivot_wider(names_from = `Next Dose`, values_from = `Follow up data` ) %>% mutate(Outcome = c('N'), `1` = '-') %>%
select(Outcome, `1`, `2`, `3`) %>%
rbind(c('T', 'Always', '-', '-' )) %>%
arrange(`2`), align = 'lccc') %>%
kable_styling(full_width = F, position = 'left') %>%
add_header_above(c(" " = 1, "Recommended Dose" = 3))| Outcome | 1 | 2 | 3 |
|---|---|---|---|
| T | Always |
|
|
| N |
|
1-30 | 31-56 |
3 Case of 2 Patients or Cohorts of 2
Calculating the next pathway becomes more tricky depending on the outcome. It is somewhat similar to having a cohort of 2 patients. In both these cases the possible outcomes are:
- TT
- NT (note NT and TN are the same)
- NN
As before the TT case is quite simple to solve as the weights/follow-up are set at 1. Similarly NT can be summarised in a similar way as above except when we run the CRM there will be an extra patient with a toxicity.
3.1 If TT Occurs
level <- c(1,1)
tox <- c(1,1)
followup <- c(56,56)
mod <- titecrm(prior = prior, target = target, tox = tox, level = level,
obswin = obswin, scheme = 'linear', followup = followup)
mod## Today: Wed Jun 23 11:18:58 2021
## DATA SUMMARY (TITE-CRM)
## PID Level Toxicity f/u Weight Included
## 1 1 1 56 1 1
## 2 1 1 56 1 1
##
## Toxicity probability update (with 90 percent probability interval):
## Level Prior n total.wts total.tox Ptox LoLmt UpLmt
## 1 0.1 2 2 2 0.629 0.205 0.873
## 2 0.2 0 0 0 0.723 0.33 0.91
## 3 0.3 0 0 0 0.785 0.436 0.932
## Next recommended dose level: 1
## Recommendation is based on a target toxicity probability of 0.2
##
## Estimation details:
## Empiric dose-toxicity model: p = dose^{exp(beta)}
## dose = 0.1 0.2 0.3
## Normal prior on beta with mean 0 and variance 1.34
## Posterior mean of beta: -1.603
## Posterior variance of beta: 0.559The reccommended dose is dose level 1.
3.2 If NT Occurs
level <- c(1,1)
tox <- c(1,0)
followupcombo <- cbind(rep(56, obswin), 1:56)
results <- data.frame('Observed' = 1:56, 'Rec' = rep(0,obswin))
for (i in 1:obswin) {
followup <- followupcombo[i,]
mod <- titecrm(prior = prior, target = target, tox = tox, level = level,
obswin = obswin, scheme = 'linear', followup = followup)
results$Rec[i] <- mod$mtd
}
kable(results %>% group_by(TD20 = Rec) %>%
summarise(n = n()), align = 'cc') %>%
kable_styling(full_width = F, position = 'left')| TD20 | n |
|---|---|
| 1 | 56 |
The recommended dose is 1 no matter how much observed data there is for the patient with no toxicity.
3.3 If NN Occurs
Here there are 3136 variations of the NN outcome. The number of variations is \((observation \; period)^ {No. \;Patients}\). We could observe both patiens for 1 day each or both for the whole window and everything inbetween.
level <- c(1,1)
tox <- c(0,0)
combos <- 1:obswin
combos <- expand.grid(combos, combos)
pos <- cbind(rep(0,nrow(combos)))
results <- pos[, rep(1, each=length(tox)+1)]
for (i in 1:nrow(combos)) {
followup <- as.numeric(combos[i,])
weights <- followup / obswin
mod <- titecrm(prior = prior, target = target, tox = tox, level = level,
obswin = obswin, weights = weights)
for (j in 1:ncol(results)) {
results[,j][i] <- followup[j]
results[,ncol(results)][i] <- mod$mtd
}
}
results <- data.frame(results)
colnames(results) <- c('Patient1', 'Patient2', 'TD20')
resultsScrolling through the table you can see that certain combinations of follow up times lead to different reccomendations for the TD.
3.3.1 Summarising using Total Follow-Up
One thought I had was to look at the total follow up of both patients depending on what dose recommendation the TITE-CRM made.
kable(results %>% mutate(TotalFollow = Patient1+Patient2) %>%
group_by(TD20) %>%
summarise(n = n(), min = min(TotalFollow), max = max(TotalFollow)),
align = 'cccc') %>%
kable_styling(full_width = F, position = 'left')| TD20 | n | min | max |
|---|---|---|---|
| 2 | 488 | 2 | 33 |
| 3 | 2648 | 31 | 112 |
One assumption used here is that both patients will have at least 1 day of follow-up. The table indicates that if the total follow-up time between both patients exceeds 33 the TITE-CRM will recommed dose level 3. Similarly, if total follow-up is less than 30 the model will always recommend dose level 2. However, there is a grey area, if the total follow-up is between 31 and 33 it could recommend either dose.
3.3.2 Visualisation of the problem
results %>%
ggplot(aes(x = Patient1, y = Patient2, fill = as.factor(TD20))) +
geom_tile() +
scale_fill_brewer(palette = 'Paired') +
theme_bw() +
geom_abline(intercept = 30, slope = -1, col = 'red',
linetype = 'dashed') +
geom_abline(intercept = 34, slope = -1, col = 'red',
linetype = 'dashed') +
theme(panel.border = element_blank(), panel.grid.major = element_blank(),
panel.grid.minor = element_blank(),
axis.line = element_line(colour = "black")) +
labs(fill = 'TD20', x = 'Patient 1 follow-up',
y = 'Patient 2 follow-up')+
scale_x_continuous(breaks = seq(0, 60, by = 05), expand = c(0, 0))+
scale_y_continuous(breaks = seq(0, 60, by = 05), expand = c(0, 0))
I am not sure what is causing this. If the observation window were longer I would assume this becomes a singular straight line.
3.4 Summarising all outcomes
kable(data.frame(Outcomes = c('TT', 'NT', 'NN'),
`1` = c('Always', 'Always', '-'),
`2` = c('-', '-', '2-30'),
`3` = c('-', '-', '34-112'), check.names = FALSE),
align = 'lccc') %>%
kable_styling(full_width = F, position = 'left') %>%
add_header_above(c(" " = 1, "Recommended Dose" = 3))| Outcomes | 1 | 2 | 3 |
|---|---|---|---|
| TT | Always |
|
|
| NT | Always |
|
|
| NN |
|
2-30 | 34-112 |
4 With 3 Patients
Possible outcomes:
- TTT
- NTT (56 variations)
- NNT (3136 variations)
- NNN (175616 variations)
4.1 TTT, NTT & NNT Outcomes
These are just extensions of the case with two patients except an extra patient has a toxicity.
4.1.1 If TT Occurs
level <- c(1,1,1)
tox <- c(1,1,1)
followup <- c(56,56,56)
mod <- titecrm(prior = prior, target = target, tox = tox, level = level,
obswin = obswin, scheme = 'linear', followup = followup)
mod## Today: Wed Jun 23 11:19:00 2021
## DATA SUMMARY (TITE-CRM)
## PID Level Toxicity f/u Weight Included
## 1 1 1 56 1 1
## 2 1 1 56 1 1
## 3 1 1 56 1 1
##
## Toxicity probability update (with 90 percent probability interval):
## Level Prior n total.wts total.tox Ptox LoLmt UpLmt
## 1 0.1 3 3 3 0.695 0.307 0.894
## 2 0.2 0 0 0 0.776 0.438 0.925
## 3 0.3 0 0 0 0.827 0.54 0.943
## Next recommended dose level: 1
## Recommendation is based on a target toxicity probability of 0.2
##
## Estimation details:
## Empiric dose-toxicity model: p = dose^{exp(beta)}
## dose = 0.1 0.2 0.3
## Normal prior on beta with mean 0 and variance 1.34
## Posterior mean of beta: -1.846
## Posterior variance of beta: 0.513The reccommended dose is dose level 1.
4.1.2 If NTT Occurs
level <- c(1,1,1)
tox <- c(0,1,1)
followupcombo <- cbind(1:56, rep(56, obswin), rep(56, obswin))
results <- data.frame('Observed' = 1:56, 'Rec' = rep(0,obswin))
for (i in 1:obswin) {
followup <- followupcombo[i,]
mod <- titecrm(prior = prior, target = target, tox = tox, level = level,
obswin = obswin, scheme = 'linear', followup = followup)
results$Rec[i] <- mod$mtd
}
kable(results %>% group_by(TD20 = Rec) %>%
summarise(n=n()), align = 'cc') %>%
kable_styling(full_width = F, position = 'left')| TD20 | n |
|---|---|
| 1 | 56 |
Similar to before the recommended dose will always be 1 no matter how much data is observed for the patient without a toxicity.
4.1.3 If NNT Occurs
level <- c(1,1,1)
tox <- c(1,0,0)
combos <- 1:obswin
combos <- expand.grid(combos, combos)
combos <- cbind(56, combos)
pos <- cbind(rep(0,nrow(combos)))
results <- pos[, rep(1, each=3)]
for (i in 1:nrow(combos)) {
followup <- as.numeric(combos[i,])
weights <- followup / obswin
mod <- titecrm(prior = prior, target = target, tox = tox, level = level,
obswin = obswin, weights = weights)
for (j in 1:ncol(results)) {
results[,j][i] <- followup[j]
results[,ncol(results)][i] <- mod$mtd
}
}
results <- data.frame(results)
colnames(results) <- c('Patient1', 'Patient2', 'TD20')
kable(results %>% group_by(TD20) %>%
summarise(n=n()), align = 'cc') %>%
kable_styling(full_width = F, position = 'left')| TD20 | n |
|---|---|
| 1 | 3136 |
With one toxicity the model still won’t escalate even if two other patients have been fully observed without tox.
4.2 NNN Outcome
I ran these separately. Took about 1 hour for all iterations.
load('TITE_DTPs_3.RData')
results <- data.frame(results)
colnames(results) <- c('Patient1', 'Patient2', 'Patient3', 'TD20')
resultskable(results %>%
mutate(TotalFollow = Patient1+Patient2+Patient3) %>%
group_by(TD20) %>%
summarise(n = n(), min = min(TotalFollow), max = max(TotalFollow)),
align = 'cccc') %>%
kable_styling(full_width = F, position = 'left')| TD20 | n | min | max |
|---|---|---|---|
| 2 | 5339 | 3 | 34 |
| 3 | 170277 | 31 | 168 |
These results can be interpreted similar to the ones before except now the total follow-up is for three patients. So, if the total follow-up is between 3 and 30 the model will definitely recommend dose level 2. If the total follow up is between 35 and 168 the model will definitely recommend dose level 3. If the total follow up is between 31 and 34 it could be either 2 or 3.
4.2.1 Visualisation
Interactive plot. Essentially a 3D version of the plot before. We could define two planes where everything above and below would be a certain dose. My linear algebra is a bit rusty so i’ll pursue this if you think its worth the time.
plot3d(x = results$Patient1, y = results$Patient2, z = results$Patient3,
col = as.factor(results$TD20), xlab = 'Patient 1 follow-up',
ylab = 'Patient 2 follow-up', zlab = 'Patient 3 follow-up')